= Dunford-Pettis theorem
{c}
On a finite-measure space, a bounded subset of $L^1$ is relatively weakly compact exactly when it is <uniformly integrable>. In particular every <uniformly integrable> bounded sequence has a weakly convergent subsequence. This is the <compactness> substitute for reflexivity at exponent one. A superlinear integral bound $\sup_j\int H(u_j)<\infty$, with $H\geq0$ and $H(t)/|t|\to\infty$, gives the required <uniform integrability>: for $H(t)\geq A|t|$ outside $[-R_A,R_A]$,
$$
\int_E|u_j|\leq R_A|E|+A^{-1}\sup_j\int H(u_j).
$$
Choose $A$ large and then the measure of $E$ small. This is the direction used in the <direct method in the calculus of variations>; a mere bounded $L^1$ <norm> does not suffice, as concentrating spikes show.
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